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Booklet 3. Discrete and Continuous Random Variables
Exercise 1 Let X be a random variable with the following probability mass function:
xi 4. 5 4. 6 4. 7 4. 8 4. 9 5. 0 P (X = xi) 0. 16 0. 25 0. 28 0. 18 0. 10 0. 03
(a) Calculate the following probabilities: P (X = 4.8), P (X > 4 .8), P (X < 4 .8) , P (X ≥ 4. 6 | X < 4 .9).
(b) Calculate the expectation and the standard deviation of X.
Ans: a) 0.18, 0.13, 0. 69, 0.8161; b) μX = E[X] = 4.69, σX = 0.1315.
Exercise 2 Calculate the mean, variance and standard deviation of a random variable X whose distribution function is:
F (x) =
0 if x < 1 , 1 / 6 if 1 ≤ x < 3 , 1 / 2 if 3 ≤ x < 6 , 1 if x ≥ 6.
Ans: mean 4.17; variance 3.80; standard deviation 1.95.
Exercise 3 An engineer wants to estimate the total cost of a project to make an offer for it. He valuate the work in terms of a fixed component of 12, 000 euros and a variable component of 300 euros per day worked. She knows the project will take between 7 and 11 days, with the following probability mass function for X = ”number of days needed for the project”:
xi 7 8 9 10 11 P (X = xi) 0.10 0.20 0.30 0.30 0.
(a) What is the probability of the project taking 9 or 10 days?
(b) Calculate the expected value and variance of X.
(c) Determine the expected cost of the project and the corresponding standard deviation.
Ans: a) 0.6; b) E[X] = 9.1 days, σ^2 X = 1.29 days^2 ; c) 14730 euros, 340.73 euros.
Exercise 4 In a computer parts factory, it is known that 6% of the parts produced by a certain machine are defective.
(a) A part is selected at random. Let X be a variable such that X = 1 if the part is defective and 0 otherwise. Determine the mean and standard deviation for this variable.
(b) If the machine produces 20 parts a day, what is the probability that none will be defective? Calculate the probability that there is at most 1 defective part. Determine the mean and the standard deviation for the number of defective parts produced daily by that machine.
(c) Assuming the machine can make as many parts as necessary, calculate the probability that the first defective part will be the fourth part. What is the expected number of parts that will be manufactured until a defective part occurs?
Ans: a) E[X]=0.06, σX =0.23749; b) 0.2901, 0.6605, 1.2, 1.0621; c) 0.0498, 17 (including the defective part).
Exercise 5 A multiple-choice test has 10 questions with 4 possible answers, only one of which is correct. If a student answers the exam at random, calculate:
(a) The probability of getting 5 questions right.
(b) The probability of not getting any question right.
(c) The probability of getting 8 questions right.
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(d) What is the expected value and standard deviation of the number of correct answers?
Ans: a) 0.0584; b) 0.0563; c) 0.0004; d) E[X]=2.5, σX =1.
Exercise 6 A customs official decides to inspect 3 of 16 boats from country P. If the selection is random and 5 of the boats contain contraband, determine the probabilities that the customs official:
(a) finds no boat with contraband.
(b) finds exactly one boat with contraband.
(c) finds exactly five boats with contraband.
Ans: a) 0.2946; b) 0.49107; c) 0.
Exercise 7 In a population it was found that the mean annual number of deaths from a given disease is 3. Assume that the number of annual deaths from the disease follows a Poisson distribution.
(a) What is the probability of no deaths from this disease in the next year?
(b) What is the probability of the number of deaths due to the disease exceeding the expected number in the next year?
(c) What is the probability of more than 9 deaths from this disease in the next 2 years?
Ans: a) 0.04979; b) 0.35277; c) 0.
Exercise 8 The manager of a post office needs to send 15 packages: 6 by airmail and the rest by surface mail. However, in an oversight she mixes them up and sends 6 randomly selected packages by airmail. What is the probability that the 6 packages sent by airmail include only 3 of the packages that should have been sent by airmail? Ans: 0.
Exercise 9 The battery life (years) of a mobile phone model is a continuous random variable with density function:
f (x) =
k(x − 2)^2 if 2 ≤ x ≤ 4 , 0 otherwise
(a) Calculate the value of k for f to be a density function.
(b) What is the probability that a battery lasts more than three and a half years?
(c) If a battery is more than three years and a half old, calculate the probability that it lasts less than 3. years.
(d) Calculate the mean and the median of the battery life. Compare their values. Calculate the 0.9-quantile of the battery life and interpret its value.
Ans: a) 3/8; b) 0.5781; c) 0.4291; d) mean: 3.5 years, median: 3.58 years, 0.9-quantile: 3.93 years.
Exercise 10 A boiler is supplied with oil once a week. Weekly consumption X (in thousands of litres) is represented by the following density function:
f (x) =
3 / 2 − x if 0 < x < 1 0 otherwise
(a) What approximate boiler tank capacity (thousands of litres) is necessary for the probability that the oil runs out in a given week to be 0.10?
(b) Calculate the expected weekly consumption.
Ans: a) 0.8292, b) 5/12 thousand of litres
Exercise 11 The reaction time (in seconds, s) to a certain stimulus is a continuous random variable with density function:
f (x) =
- 5 x^2 if 1^ ≤^ x <^3 0 otherwise
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Ans: a) 0.4493; b) 0.53 years.
Exercise 16 A company mounts computer spare parts. The company employees work independently of each other and the number of parts mounted by each employee follows a Poisson distribution with mean 24 parts per 5 hours.
(a) Calculate the probability of an employee lasting more than 45 minutes in mounting one part.
(b) Calculate the probability that two employees mount jointly more than two parts in half an hour.
(c) If the company has 100 employees, calculate the probability that the company mounts more than 2500 parts in 5 hours.
Ans: a) 0.0273; b) 0.85746; c) 0.0202.
Exercise 17 A company that produces cork stoppers has two manufacturing machines. The first machine produces stoppers with diameters following a Normal distribution with mean 3 cm and standard deviation 0. 1 cm. For the second machine, diameters follow a Normal distribution with mean 3.05 cm and standard deviation 0 .05 cm.
(a) Draw an approximate graph showing the diameter density functions of both machines.
(b) A stopper is commercially accepted whenever its diameter is between 2.9 and 3.1 cm. Which machine produces the greatest number of acceptable stoppers? Calculate the corresponding probabilities.
(c) If we were able to change the mean diameter obtained by the second machine (without changing the standard deviation), what mean value should be chosen in order to get a maximum percentage of acceptable stoppers in that machine? Justify the answer. What would be such a maximum percentage?
(d) If the stoppers produced by the first machine are packed in boxes of 10 independent units. What is the probability that a randomly chosen box contains at least one unacceptable stopper.
Ans: b) machine 1: 0.68268, machine 2: 0.83999; c) μ = 3 cm, probability 95.45%; d) 0. 9780
Exercise 18 The net weight of the sardine cans produced by a cannery follows a Normal distribution with mean 250 gr and standard deviation 5 gr.
(a) For a can to be considered acceptable, its net weight should be between 242 and 258 gr. What is the proportion of acceptable cans?
(b) What is the minimum weight of 10% of the cans with highest net weight?
(c) What is the probability that in a batch of 20 cans at most one is not acceptable?
(d) What is the probability that the total net weight of a batch of 100 cans will be greater than 25.1 kg?
Ans: a) 0.8904; b) 256.4; c) 0.3396; d) 0. 02275
Exercise 19 The start of an office working day is 9:00 am. The time at which a certain employee arrives at work follows a normal distribution with mean 9.1 h and standard deviation 0.05 h, depending on traffic on the way to work.
(a) Calculate the probability of late arrival.
(b) Assuming that one day he is late, what is the probability that he is late by at most 15 minutes?
(c) If his boss observes him for 5 days, what is the probability of his arriving late on at least 3 days?
Ans: a) 0.97725; b) 0.9986; c) 1 approx.
Exercise 20 Let X follow a distribution N (μ, σ), calculate the following probabilities:
(a) P (X < μ + 2σ)
(b) P (|X − μ| < σ)
(c) P (|X − μ| < 2 σ)
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(d) P (|X − μ| < 3 σ)
Ans: a) 0.97725; b) 0.68269; c) 0.95450; d) 0.99730.
Exercise 21 The daily demand (litres) for a brand of milk is a random variable X with distribution N (3000, 350).
(a) What minimum daily production would be necessary to meet total demand at least 90% of the time?
(b) Calculate the probability that total demand for 100 days exceeds 307000 litres.
Ans: a) 3448.543 litres; b) 0.02275.
Exercise 22 A manufacturer knows that 20% of batches are rejected by customers.
(a) Calculate the probability that exactly 3 batches in a total of 10 batches are rejected.
(b) Calculate the probability that 80 or more batches in a total of 500 batches are rejected.
Ans: a) 0.2013; b) 0.98899 approx.
Exercise 23 The number of misprints per page of a book follows a Poisson distribution with a mean of 0.75.
(a) What is the probability of a page having more than one error?
(b) What is the probability of fewer than two errors in two pages?
(c) Calculate the probability of fewer than 350 errors in 500 pages.
Ans: a) 0.1734; b) 0.5578; c) 0.09342 approx.
Exercise 24 The Rockwell hardness of a metal is determined by measuring depth of penetration of an indenter. Assuming that the Rockwell hardness of a certain alloy follows an N (70, 3) distribution, calculate the following:
(a) If the acceptable range of hardness is (70 − c, 70 + c), for what value of c will 95% of cases have acceptable hardness values?
(b) Observed independently are 10 specimens of this alloy. What is the probability that at most 3 have a hardness greater than 73.84?
(c) Calculate the probability of having to choose more than 3 specimens before finding one with a hardness value below 73.84.
Ans: a) c = 5.88; b) 0.9872; c) 0.001.
Exercise 25 A certain machine should cut metal bearings with a diameter of 0.5 cm. In practice several factors affect the manufacturing process. Therefore, the diameter of the bearings is a random variable X = 0 .505 + 0. 01 Z, where Z is a random variable with distribution N (0, 1).
(a) What percentage of bearings produced by this machine have a diameter greater than 0.52 cm?
(b) For a batch of 10 bearings, what is the probability of any exceeding 0.52 cm in diameter?
(c) For a batch of 1000 bearings, what is the probability that less than 60 will exceed 0.52 cm in diameter?
(d) What percentage of bearings produced by this machine have a diameter which is within 0.01 cm of the nominal diameter (0.5 cm)?
Ans: a) 0.0668; b) 0.4992; c) 0.1762; d) 0.6247.
Exercise 26 A game of Tetris for Android produces small squares, randomly and independently. The length (mm) of one side of each of these squares is a random variable X with xi = 1, 2 , 3 , 6 possible values and the probability mass function P (X = xi) = xi/12, i = 1,... , 4.
(a) Determine the expected value of the area of these squares.
(b) Of 10 squares produced in the game, what is the probability of at least 2 squares having an area under 9 mm2?
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(d) The lamps are packed for sale in batches of 20 units. What is the probability that a batch does not contain any defective lamp?
Ans: a) 0.04550; b) 2082.25; c) 8000, 100; d) 0.3940.
Exercise 31 A factory produces 200 units per month of an electric motor. The lifetime of the motor follows an exponential distribution with a mean of 50000 hours.
(a) What is the probability that a motor lifetime exceeds the mean?
(b) The manufacturer warranty needs to indicates a motor lifetime value that covers 90% of motors (in other words, only 10% of motors may fail to last this long). What is the value that should be indicated as the maximum warranty period?
(c) Knowing that an engine has exceeded 60000 hours, what is the probability of failure before 70000 hours?
(d) Calculate the probability that the average lifetime of all motors manufactured in a given month exceeds 55000 hours (assume independence in the motors lifetimes).
Ans: a) 0.3679; b) 5268 hours; c) 0.1813; d) 0.07927.
Exercise 32 The length (cm) of the parts produced in a factory is a random variable X with the following distribution function:
F (x) =
0 if x < 1 (x − 1)^2 if 1 ≤ x ≤ 2 1 if x ≥ 2
(a) What percentage of parts are between 1.25 and 1.75 cm long?
(b) Determine the first quartile of the X variable and interpret.
(c) Calculate the mean of the random variable X.
(d) If we have 200 parts, calculate the probability of at least 60 parts measuring less than 1.5 cm.
Ans: a) 0.5; b) 1.5 cm; c) 5/3 = 1.6667 cm; d) P (Z ≥ 1 .33) = 0.06057 approx.
Exercise 33 Let X be the random variable representing the time (hours) that the battery of an electronic device functions properly between consecutive recharges. The density function of X is given by:
f (x) =
3 x
− (^3) if 20 ≤ x ≤ 40 0 otherwise.
(a) Calculate the average duration of the battery charge before recharging is required.
(b) Calculate the probability of the battery running out within 24 hours after a recharge.
(c) Calculate the probability of the battery running out within 36 hours after a recharge, if you know that it is still working after 24 hours.
(d) If recharges are independent, what is the probability that the fourth recharge is the first charge lasting less than 24 hours?
Ans: a) E(X) = 80/3 = 26.67 h; b) 0.4074; c) 0.8681; d) 0.0848.
Exercise 34 It is decided to install underwater turbines in an area where 95% of the time current flow is between 3 and 10 km/h. It is known that the current flow is a random normally distributed variable with mean 6 .5 km/h.
(a) Calculate the standard deviation.
(b) A turbine can only be used when current flow is less than 10 km/h. What is the probability of not being able to use a turbine?
(c) If 12 turbines of this type are installed that operate independently, what is the probability that more than 10 turbines can be used?
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(d) What is the first quartile of the current flow?
(e) If 100 measurements of current flow are recorded, what is the probability of the sample mean being between the values 6.25 and 6.75 km/h?
Ans: a) 1.786; b) 0.025; c) 0.9651; d) 5.295; e) 0.83848.
Exercise 35 An electrical installation can use two types of spotlights, A and B. The duration (hours) of type A and B spotlights is a random variable with distribution N (800, 100) and distribution N (900, 150), respectively.
(a) What percentage of spotlights A last longer than 900 hours?
(b) What duration is exceeded by only 10% of spotlights B?
(c) Two spotlights, one type A and the other type B, are installed at the same time. Assuming both spotlights operate independently, what is the probability that type A will last longer than type B?
Ans: a) 0.15866; b) 1092 horas; c) 0.29116.
Exercise 36 The number of bumps of a very deteriorated county road follows a Poisson distribution of average 15 bumps per kilometre. Calculate the probability that:
(a) There are less than 4 bumps in a 500-metres section.
(b) There are no bumps in a 100-metres section.
(c) The distance to the next bump is between 100 and 200 metres.
Ans: a) 0.0591; b) 0.2231; c) 0.1733.
Exercise 37 Totis, an elevator company, wishes to determine the weight to be supported by its new elevator model so that the probability of failure due to excess weight is 0.1. The elevator is designed to accommodate a maximum of 9 people on each trip and tests are done with that number of people. Previous studies of the population of users indicate that their weight is a normal random variable with mean 75 kg and standard deviation 10 kg.
(a) Calculate the maximum allowable weight for a 0.1 probability of failure due to excess weight for 9 people using the elevator (assume that the weights of the 9 people are independent random variables).
(b) Assuming that the probability of failure due to excess weight is 0.1, what is the probability that, after 20 trips, the elevator fails at least once due to excess weight?
(c) Assuming that the probability of failure due to excess weight is 0.1, on average, how many trips will take place before the elevator fails due to excess weight for the first time?
Ans: a) 713.4 kg; b) 0.8784; c) 9 trips (10 including the trip of the fail).
Exercise 38 The time (days) that customers delay in paying a company is a continuous random variable X, with the following distribution function:
F (x) =
0 if x < 10 , 1 − 100 x−^2 if x ≥ 10.
(a) Calculate the probability of a customer paying within 15 days.
(b) What is the median time that customers delay in paying?
(c) The company is especially interested in customers that delay more than 15 days in paying. Calculate the probability of one of these customers paying within 20 days.
Ans: a) 0.5556; b) 14.14 days; c) 0.4375.
Exercise 39 The time (minutes) that a laboratory technician needs to prepare an equipment is a random variable X with a uniform distribution on the interval (20, 30).
(a) Calculate the mean and the standard deviation of X.
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Excel 2010: To generate data from a variable with a particular distribution, this option must be enabled in Excel. To do this, access the menu Archivo + Opciones + Complementos + Administrar: Complementos de Excel IR + Herramientas para el an´alisis VBA. Once activated we can generate random numbers in Datos + An´alisis de datos + Generaci´on de n´umeros aleatorios. Indicate the number of variables to be generated, the number of random numbers required, the distribution of the variable (specify its parameters), and finally, the initial position for the random numbers generated.
(b) Calculate and graphically compare the probability mass function and the distribution function for the random variable Bi(8, 0 .4). Is there symmetry in the probability mass function?
Excel 2010: The function DISTR.BINOM.N() calculates the probability mass function of a ran- dom variable with a binomial distribution. The syntax of this function is:
DISTR.BINOM.N(n´um ´exito; ensayos; prob ´exito; acumulado)
with n´um ´exito (x): the value for which we want to evaluate the probability function. ensayos (n): the number of experiment realizations. prob ´exito (p): the probability of success in each realization. acumulado: a logical value. If acumulado=0 (FALSE) the probability of x successes is obtained, and if acumulado=1 (TRUE) the distribution function in x is obtained.
(c) Graphically compare the functions obtained with the data simulated in (a1) above with the corresponding theoretical functions obtained in (b).
Exercise 44 Normal Distribution in Excel.
(a) Calculate and plot the density function f and the distribution function F of two random variables with distributions N (0, 1) and N (1, 0 .5) for values ranging from −3 to 3 in increments of 0.1.
Excel 2010: The function DISTR.NORM.N() calculates the density function or the distribution function of a random variable with a normal distribution. The syntax of this function is:
DISTR.NORM.N(x; media; desv est´andar ; acumulado)
with x : the value where we want to obtain the density function or the distribution function. media (μ): mean of the variable. desv est´andar (σ): standard deviation of the variable. acumulado: a logical value. If acumulado=0 (FALSE), the density function is obtained, and if acumulado=1 (TRUE) the distribution function is obtained.
(b) For X d ∼ N (1, 2), calculate the following probabilities: P (X ≥ 1), P (X ≥ 1 .5), P (X ≥ 4), P (X ≤ 0 .5) and P (− 1 ≤ X ≤ 0 .5). Justify that P (X ≥ 1) = 0.5 and P (X ≥ 1 .5) = P (X ≤ 0 .5).
(c) For a distribuci´on N (10, 3) calculate the quantiles 0.01, 0.025, 0.05, 0.95, 0.975 and 0.99. What relationship exists in normal distributions between the quantile p and the cuantil 1 − p? For example, compare the 0 .05 and 0.95 quantiles. Calculate these same quantiles for the standardized variable Z d ∼ N (0, 1). What relationship exists between the quantiles of the distributions N (10, 3) and N (0, 1)?
Excel 2010: The INV.NORM() function calculates the quantiles of a normal distribution. The syntax of this function is:
INV.NORM(p; media; desv est´andar)
with probabilidad (p): the probability giving rise to the corresponding quantile. media (μ): mean of the variable. desv est´andar (σ): standard deviation of the variable.
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Exercise 45 Verification of the Central Limit Theorem through Excel simulation. We will empirically or experimentally verify that the sum of a large number of random variables, independent and identically distributed, approximately follows a normal distribution. The steps are as follows:
(a) Generate 10 random variables X 1 ,... , X 10 independent and identically distributed, with distribution U [0, 1] using the Excel ALEATORIO() function. Repeat this step until obtaining 5000 values for each variable.
(b) Calculate the 5000 values of the variable sum S 10 = X 1 + · · · + X 10.
(c) Approximate the density function of S 10 using a histogram. Does this histogram look like the histogram of a normal distribution? What values should have the parameters of a normal distribution for approximating the distribution of S 10?
Note: The number of intervals K can be calculated using the so-called Sturges” rule:
number of intervals = 1 + log 2 (n) = 1 + 3.322 log 10 (n),
where n is the number of observations. The length of the interval is calculated using the rule
L = (Maximum Datum - Minimum Datum)/K.
A similar analysis can be done for the sum of binomial, normal, exponential, etc, independent variables.
